Engineering and Architecture /5. Heat and Gas Supply and Ventilation

 

Vladimir I. Sokolov, Alim A. Kovalenko

 

Volodymyr Dahl East-Ukrainian National University, Ukraine,

 

Inna K. Nasonkina

 

Donbass National Academy of Civil Engineering and Architecture, Ukraine

 

The Stages of Aerosols’ Diffusion in Turbulent Flows

The estimation of diffusion process characteristics of aerosols in the ventilation systems of industrial enterprises requires the research of discrete particles’ behavior in a turbulent flow. Such behavior depends on the concentration of these particles and their size, on comparison to the scale of turbulence working media[1-3]. This is a pressing issue at measuring of aerosols’ concentration in the exhaust of nuclear power plants.

When concentration is high the collision of particles and the influence of media in the vicinity of particles cause the direct interaction of particles. In case when concentration of particle is not high the interaction between the particles can be neglected and each particle is considered as it is the only in the turbulent flow. This is very common in practice.

There are the logical assumptions for gas and aerosol exhaust in the channels of vent systems:

· turbulence in a flow is homogeneous and stationary;

· particle is spherical and it is so small that it motion relatively of media submits to the Stokes law resistance;

· particle is very small in comparison to the smallest length of the wave in turbulent motion;

· any external force which influences on a particle is related to a potential field, for example gravity.

Recognizing these assumptions the behavior of spherical particle in a flow can be described with equation of Basse [4]:

                        ,                        (1)         

where          

                    ;                   (2)      

t - time; t0 - an initial moment of time; index f - related to basic media, and index p - to a particle; m - a dynamic viscosity of media; d - a diameter of particle; rp and rf are densities of particles and gas stream; vp - a speed of a discrete particle; vf - a speed of particles of basic media in vicinity of discrete particle far enough not to test indignations of relative motion of this particle.

The coefficients of temporal Lagrange correlation for the interval of time  t are used as characteristics of turbulent motion

                      ,                     (3)

and the Lagrange functions of power spectrum are

                              ,.                            (4)          

Here  are the pulsation components of speeds of discrete particle and gas flow and an index <…> designates that pulsating value in turbulent flow becomes a mean value.

It  be mentioned that coefficients of temporal Lagrange correlation is related to the Lagrange functions of power spectrum with the help of relation

 

    ,    (5)         

We must add that the coefficient of the Lagrange correlation of basic flow is usually presented as close to exponential dependence for implementation of quantitative estimations [4]

                                             ,                                            (6)

where is the Lagrange integral temporary scale taken as a measure of the longest time interval during which a particle moves in a given direction .

Linking the Lagrange variables we write down the relations for the Lagrange coordinates with the help of the function of power spectrum

          ,.         (7)

As the coefficient of turbulent diffusion is , then we obtain the following for a discrete particle and gas

                                                   .                           (8)            

Under small time of diffusion

                      , ,

from where there is

                                        .                                       (9)

Under long time of diffusive process a main role belongs to the low frequency components of motion

             (10)

and

                                                .                                             (11)

 According to expressions (10) and (11) the coefficient of diffusion is proportional to the part of kinetic energy in turbulent motion with a zero frequency. But there is no difference between the motion of a particle and the motion of media under condition of zero frequency. Thus, it is very logical from physical point of view that coefficients of diffusion for a discrete particle and particles of basic stream must be the same. Therefore

                                               .                                            (12)

We must underline that those notions of “small” and “long” time of diffusion are very relative and must be defined in every particular case. The expresses (12) is for infinitely large time of diffusion. As the channels of real systems have boundaries, in most of cases there will be the difference in values of coefficients of turbulent diffusion.

We estimate the duration of induction period where the Stokes forces cause an acceleration of particles to speed of basic flow. For this purpose in equation (1) we will ignore the last term. Such assumption can be taken as fully legitimate because forces of resistance and forces related to the gradient of pressure and to the relative acceleration of mass become vital. Then we obtain

                                        .                                      (13)

In this case the constant coefficients are defined according to equation (2).  As for aerosol exhaust in air flow the value has an order 10-3  and we can establish that b»0. That is why the equation (13) can be presented as

                                                                                             (14)       

We integrate it under initial condition that  for gas flow speed. There is a solution (14)

                                           ,                         

from where we obtain

                            .                         (15)

And 

                                          .

Thus, recognizing (15) from expression (9) we obtain

             .           (16)

According to (16) we estimate the time of induction period. If the end of induction period is counted by the value , taken in most of technical calculations, then the time of period is  . Taking to consideration the expression (2) we have  for coefficient a and taking into account that has an order 10-3  we establish the following relation for the time of induction period

                                                   .                                                 (17)

After the induction period is over the power spectrums of particles and of basic flow are not the same therefore the diffusive coefficients are different. Thus, the diffusive period under variable coefficient of diffusion of aerosol particle will take place. Taking the coefficient of the Lagrange correlation for gas flow as exponential dependence (6) we obtain the corresponding power spectrum in dependence on frequency of pulsations w.

                                        .

Considering equation (13) the power spectrum for discrete particles is

                               .

Then

    

and according to (5)

                                 

                  

By the coefficient of the Lagrange correlation we define the coefficient of diffusion ((3),(4) and (8)).

As there is

                                          (18)

then we obtain

                           .                               

As mentioned above << 1 then we are supposed b » 0 and . From here we obtain

                           .                         (19)  

Let us estimate the Lagrange integral scale of time. Based on (18) where t®¥

                                               ,

and taking to consideration that degree of turbulence is , we obtain

                                                    .                                                 (20)

where u0 is an average speed of flow in a channel.

The following transformations are done

                               ,

where  is the Reynolds number; is kinematics viscosity of basic flow, dã  - a hydraulic diameter of channel; is a scale of time equal to the time of passage of a basic flow particle through the way which is equal to hydraulic diameter of channel.

Examining the Peclet number     

,

we obtain

                                                  .                                                (21)

The calculations have shown that for a degree of turbulence e = 0,05 in diapason of the Peclet diffusion numbers 300...400 considered for typical of ventilation the Lagrange value of integral scale is . Thus, it approaches the time -of passing of a basic flow particle through the way equal to hydraulic diameter of channel.

It substitute (21) to (20) and put into operation dimensionless time .

                      

Considering that  ;  è ,

              .            (22)

As it is seen when   it is completely correspond to the result (12).

According to (22) maximum dimensionless time of the transitional period in a diffusive process under condition that aerosol particle’s coefficient of diffusion approaches the basic flow coefficient of diffusion is

                             .                         (23)

This relation shows that time of transitional period depends on different factors and it must be estimated in every particular case.

Thus, the process of aerosols’ diffusion in turbulent flow is very compeck and can be divided into 2 initial parts: the induction period, when the Stokes forces cause an acceleration of particles to speed of basic flow, and the transitional period, when the process of diffusion takes place under variable coefficient of diffusion. The diffusive process in the induction period is almost absent and the duration of this period is determined by equation (17). Most of  transitional period is set by relation (23) and the coefficient of aerosols’ diffusion is estimated by relation (22).

Literature

1.                     Rutherford A. Introduction  to  the  Analysis of Chemical  Reactors. Departments  of Chemical Engineering University of Minnesota. Prentice - Hall. Inc., New Jersey, 1967. - 340 p.

2.                     Ñîêîëîâ Â.È. Àýðîäèíàìèêà ãàçîâûõ ïîòîêîâ â êàíàëàõ ñëîæíûõ âåíòèëÿöèîííûõ ñèñòåì/ Â.È. Ñîêîëîâ. – Ëóãàíñê: ÂÓÃÓ, 1999. – 200 ñ.

3.                     Õèíöå È.Î. Òóðáóëåíòíîñòü/ È.Î.Õèíöå. - Ì.: Ôèçìàòãèç, 1963. - 680 ñ.

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